Archive for February, 2012

Further Understanding e and Exponential Functions

Brett
Feb 27 |11:52

You know, we had a lot of fun today.  From understanding the bases of exponential functions, to using our calculators to zoom in on a parabola, today sure was a blast.

But what’s even more important than having a ball in math class is to retain that information for future use.  That’s the real fun, isn’t it?  So without further ado, I present to you, The Blog.

So we started off with a couple of warm up problems.  The SmartBoard told us to make 81^5 equal to 3^20 without actually calculating the actual values that these produce.  It also told us that by “manipulating” either one or both sides of the equation, we could make these sides equal.  The first thing that I recognized was that 81=3^4 and that I could use this to help further down the line.  If i could reduce the original equation down to this, it would be solved.

What was required to do next was to simply substitute the 81 and 3^4 into the original equation.  This gives us (81)^5 on the left, and (3^4)^5 on the right to keep it equivalent to the original 3^20.  If you take it one step further you can take the fifth root of both sides and get it to the equivalent that we knew previous to this problem.  This leaves us with 81=3^4, which we all know to be true.  Knowing this skill can now help us in future situations to simplify exponents, and even replace bases with other bases.

Speaking of replacing bases, that was our next subject.  We used the same exact situation to learn this.  The board read “81^x=3^?”.  In other words, if you want to write one exponential value that has 81 for a base, how can you write an equivalent exponential value with 3 for a base.  This was actually quite simple to figure out, as all we had to do was look at the power of 81 in the first problem (5) and easily relate it to the power of 3 (20).  Obviously, you can just multiply 81′s power by 4 to get to 3′s power, which leaves us with a solution: 81^x=3^4x.  The reverse of that is what we solved for next in the same way which got us 3^x=81^(x/4).

To further reinforce this concept, we did the same thing with 7 and 5.  This was also quite simple if you understood the topic, as all we really need to do is plug in 1 for x in the equation 7^x=5^? to get us 7=5^x.  From that point on, it’s really just a whole lot of guessing and testing on our calculators.  In class we found that x was approximately 1.21, but to be a bit more precise, it was closer to about 1.2090619551.  Again, we did the opposite with this, 5=7^x.  To acquire x, all we really need to do is find the reciprocal of 1.21 like we did in the first one with 4x to get 1/4x.  We found our final solution to be 5^x=7^.83x.

From these challenge problems came the basis of our lecture for the day, which really told us what we were to take away from today’s class: ANY exponential value can be written under ANY other base by changing the power of the original exponential value.

We also reviewed how we can find the slope on an exponential function give any point on the graph.  To show this,  we got out our handy dandy TI-81′s and graphed a simple parabola.  After zooming in a countless number of times on the vertex of it, we found that at the very bottom of it, we have a straight line that has a slope of zero.  Another way of thinking about this is that on the left side, the parabola’s slope is negative, but slowly approaching 0 until it finally does at the vertex and then becomes positive.

Well that’s all folks, hope you enjoyed and learned lots from it!

 

Oh yeah, and I promised Dundas a sweet dominoes video, but I decide to mix things up with something a little different.

OK Go -This Too Shall Pass – Rube Goldberg

 

Understanding e

Reversearp
Feb 21 |09:41

Please read and then comment (here) on this blog post:  http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/

 

Multiply-Multiply and Add-Add Properties

LeahA
Feb 13 |17:04

Today we expanded our knowledge on Power Functions, by learning about a specific property that Power Functions have, the Multiply-Multiply Property.

As a reminder, a Power Function always follows the following format:  f(x)=a?x^b          

The property that was given to us states that:  If f is a power function, multiplying x by a constant results in multiplying f(x) by a constant.  Next, we came up with equations that helped to illustrate this property.

if  f(x)=a?x^b      x2=c?x1       then…

f(x2 )=c^b?f(x1)

 In the equations above, c is a constant,  x1 represents the first step in a function, and x2 represents the second step in a function. (The 1 and 2 should be subscripts!!!) 

Here is an example we did in class: 

We first had to find the constant (c) in which all the x values were multiplied by.  It turned out to be 2, so we knew that c=2.  Next, we found that the outputs of the function were also multiplied by a constant value, which in this example was 1.5.  Because both the inputs and outputs are multiplied by a constant to find the next value, we knew that it followed the Multiply-Multiply Property!  Our next goal was to find out what b was.  To do this, we plugged values into the equation above (  f(x2 )=c^b?f(x1)  ).  x2=18 because it is the second output of the function and x1=12 because it is the first input of the function.  We could then solve for b.  Once you get to the point where b is the exponent, you have to use the guess and check method or the natural log button on your calculator. 

The next property we illustrated during class was the Add-Add Property.  This property is very similar to the Multipy-Multipy property, however instead of multiplying the inputs and outputs of a function by a constant value, you add a constant value to each.  This property applies to Linear Functions, which follows the format f(x)=ax+b.  Here is the example we did in class to show this property:

Because a constant value is added to both the inputs and outputs of this function to get the next value, it follows the Add-Add Property and is a Linear Function.

That’s about it for today’s class!  I hope this helped to reinforce what we learned today :)

 

Power Functions

Jess
Feb 10 |01:37

In the wise words of Tao Te Ching, “Mastering others is strength. Mastering yourself is power.” However, more powerful than mastering yourself is mastering the unit we’re learning in class right now, called Power Functions!

First off, we have to understand the basics. A Power Function is defined as any function in the form y=Ax^b. In power functions, A?0 and B?0.Some examples would be X² and X³, or even X^¼ and X^¾ (assuming A=1) With that knowledge, you can begin to learn special properties that can be utilized with these functions. Let’s start with the basic behaviors of their symmetry.

Alright. By now you should be quite familiar with symmetry, but in case you need a refresher, here goes. Symmetry occurs when (in this case) one half of a graph is the “mirror image” of the other half. The line the two halves are mirrored off of is called the Axis of Symmetry. As an example, we can graph the power functions X¹,X²,X³ shown, respectively, below.

- In the graphs, the blue line represents the Axis of Symmetry. Both X and X³ have the same Axis of Symmetry over (-x,y), while X² has its across the y-axis. Based on that information we can tell that X² is an even function because they have Axises of Symmetry over the y-axis, and that X¹ and X³ have Axises of Symmetry over the (-x,y) line, which is an attribute of even functions.
.
So, Power Functions with odd exponents will result in odd functions, while Power Functions with even exponents will result in an even function. Pretty simple stuff.

-A Power Function graph can also be Symmetric with Respect to the Origin when there’s a (-x,-y) value on the graph for every (x,y) value. So a graph that’s Symmetric with Respect to the Origin would not necessarily have to pass through the origin itself as long as every (x,y) has a (-x,-y). If you look at the Power Function graphs, you can tell that X¹ and X³ are Symmetric with Respect to the Origin, whereas X² is not. This is because in order for a Power Function to be able to have a negative y output, the exponent would have to be odd. For example:
.
.
If you plug in 3 and -3 as (x) into an odd Power Function like y=1x³, you’d end up with
3·3·3=27 (3,27) and -3·(-3)·(-3)=-27 (-3,-27). Thus X³ is Symmetric with Respect to the Origin.
.
.
Now if you were to put in 3 and -3 into an even Power Function like y=1x², you’d end up with 3·3=9 (3,9) and -3·(-3)=9 (-3,9), so it would not fit the requirements to be Symmetric with Respect to the Origin.
.
.
-So, we can come to the conclusion that even Power Functions will not be Symmetric with Respect to the Origin, and odd Power Functions will.
.
.
.
-Another property of Power Functions can be found by looking at their End Behavior. Quite simply, you examine how the function “ends” by inserting ? and (-?) in for x and looking at how the y-values are affected. A Power Function’s End Behavior depends on whether the exponent (b-value) is odd or even, and if its positive or negative. First we’ll start with even Power Functions.
.
.
-Even Power Functions have even exponents. This means that they take on the shape of a parabola on a graph. As one can see by that shape, as you move farther away in either direction from the origin, the y-values would continually increase exponentially (their End Behavior).
.
.
-Odd Power Functions have odd exponents. Besides the exception of X¹, most odd Power Function’s graphs would look like that of X³ (thought End Behavior is still the same). It’s parabolic shape in the 1st quadrant would give it an End Behavior similar to that of the 1st quadrant of X², with the y-value continually increasing a bit more because it has a larger b (exponent) value. However, as you move further away from the origin towards the negative side, the End Behavior is a continually decreasing exponentially y-value. This is because is you put a negative number in for (x) in an odd function, (y) will always be negative as well.
.
.
-I’ve gone over the End Behaviors of positive (b) values, but what happens when your (b) value is negative? First, remember that X^-1 is equal to (1/X¹), or the inverse of the function with the same exponent but positive. These negative b-value Power Functions will give you an inverse graph like this:
.

X^-1 or 1/(x^1)

-This graph of x^-1 shows the inverse  of the Power Function X¹, and there are several inferences on its End Behavior we can determine by looking at its shape and the equation.
.
- As (x) gets bigger going either way from the origin, the y-value will inversely decrease, becoming smaller and smaller, yet never reaching zero. This is because in this case (x) is the denominator in a fraction and you can’t divide by zero. The function has a domain of x?0. As (x) increases the y-value will just become a smaller and smaller fraction. Also the y-value will never equal zero because nothing you could put in for (x) in 1/(X) would ever get you zero. Thus the End Behavior each way from the origin in a Power Function with a negative (b) value is an decrease in (y) becoming smaller and smaller as a fraction, never reaching zero.
.
.
.
-Also, take note that every single Power Function will have the point (1,1). This is because no matter what value you have for (b), even, odd, positive, or negative, putting 1 in for (x) will give you a y-value of 1. If you don’t believe me, just try it out! You’ll see it remains true in every case.
.
.
-A helpful tool in determining which Power Functions can be used to model data you collect is The Ladder of Powers. Basically it’s just a list of Power Functions with X¹ in the middle, and increasing b-values going up, and decreasing b-values into the negatives going down. When you have data plotted you can test out different Power Functions off  The Ladder of Powers to find the best fit line for further extrapolation.
.
.
- Power Functions that have a b-value that’s a fraction are called Rational Power Functions. Their equation would look like this: y=Ax^(m/n), where m and n are non-zero integers. This function is equal to writing it like n?(x^m) and (n?x)^m. Because of this your domain can’t have any negative numbers, because you can’t have a negative number under a Root symbol. Because of this Rational Power Functions only show up in the 1st quadrant (The 4th quadrant is never used in power functions because you will never have a positive x-value get a negative y-value). Also, know that if (b) is a fraction the function will increase in y-value, but at a very slow rate. The closer to 0 the fraction is the slower it is.
.
.
-Last but not least, let’s take a look at how the b-values and A-values affect the way our graphed Power Functions look. I mentioned a bit about it before but here’s the jist of things (with visual aid):
.

Power Functions

- On the graph the functions X^4, X³,X²,X¹,X,X^½,X^(1/3), and X^¼. In this situation the A-value is being kept at a constant 1. Notice that even though the Power Fractions with b-values less than 1 look different from the others, their y-values are still increasing, just at a much, much slower rate. You can tell that the b-value determines the shape of the curve for graphed Power Functions. 
.
.
-But, what if we changed the A-value!
.

Change in A-Values

.
-While keeping the b-value constant, we can determine the effect of a change in A-value on our Power Function Graph. In this visual to the right we are keeping (b) at a constant value of 2. Then we had A-values of 2,1,½, and -1. As you can tell, the larger the A-value was, the steeper the curve was. Also when there was a negative A-value, the parabola opened up upside down. Both of these clues are huge indicators that the A-value in a Power Function directly affects its slope.
.
.
-So, in the Power Function equation y=Ax^b, the A-values directly affect the slope while the b-values determine the shape of the curve.
.
.
.
.
.
.
Well, that just about sums up our packet and past two classes of Power Functions. I know it’s a lot of information to take in all at once but if you have any further questions about Power Functions you can check out this very helpful video:
.
Extra Power Function Help (Great Information)
.
.
TODAY’S WARM UP:
.
We were given this equation to simplify:
.    
 (2^(n+4)-2(2^n)/(2(2^n+3))
.
First off you can re-write it so that you include the 2′s ^1 on the 2s that don’t already have an exponent:
.    
(2^(n+4)-2^1(2^n)/(2^1(2^n+3))
.
Then you can multiply the 2^1′s with the numbers in the parentheses next to them to get:
.
 (2^(n+4)-2^(n+1)/(2^(n+4)) (Remember that if you had (X^(n+m))+(X^b), it would be equivalent to X^(n+m+b))
.
After you have that, you are able to split it into two separate fractions like so:
.
(2^(n+4)/2^(n+4))-(2^(n+1)/2^(n+4))
.
You can see that the first fraction is equal to one, and for the second fraction, when you divide powers you subtract the denominator’s power from the numerators, giving you:
.
1-(2^(n+1)-(n+4)
.
1-(2^-3)  (2^-3 can be written as 1/(2^3), which is equivalent to (1/8))
.
1-(1/8)
.
7/8 (ALL DONE!)
.
.
 

Power Functions!!!

Andrew
Feb 8 |00:23

Hey guys!  So for the past few days we’ve been learning about these things called #Power Functions.  It may seem confusing at first but if you #justlookatit and remember a few key details you should be just fine. 

To start off you first need to take a good look at your packet and learn a few simple definitions from your packet.  The first definition you should learn is what the heck a power function is.  A #Power Function is defined as any function in the form y=Ax^B.  In this equation A does not  equal zero and B is a non-zero interger. 

The next set of definitions that are necessary for looking at #Power Functions either on a graph or your calculator, have to deal with symmetry.  I would hope that everyone knows what symmetry is, but in case your drawing a blank it is when one half of a graph is the mirror image of the other half.  The line that acts as a mirror to that function is called the axis of symmetry.  Finally a graph that is symmetric with respect to the origin is one that for every x and y value on the graph the negative of the x and y values are also on the graph.  With this information you can start to identify some characteristics of different #Power Functions.  I hope that was just a review for you guys, but if it wasn’t then your welcome. 

Another way you can identify some characteristics about these functions is by looking at the end behavior of it.  In other words examining what happens to these functions as x becomes larger or smaller.  By graphing these functions you can get a better visual on what they look like to help you detirmine what happens to the functions as x gets bigger or smaller.  End behaviors are written like so: x (right arrow) infinitey sign (looks like a side ways 8).  This would mean that as x gets bigger the function also gets bigger or appraches infinity.  (Sorry if that was confusing, but I didn’t know how to put in an arrow or infinity sign.)  =(  

 In addition to the end behavior of #Power Functions you can also use the ladder of powers to help you see how increasing or decreasing the power of a function will affect the shape of the function.  Basically as the power increases the skinier your graph will be and as the power decreases the wider the graph will be.  If you want to see it for yourself try plugging some power functions into your calculator for example x^2 compared to x^4 and see what is different in the two graphs. 

The final piece to the puzzle of #Power Functions for now is rational Power Functions.  These are simply #Power Functions where the power x is taken to is a fraction(x^m/n) , which is equivilant to taking the nth (whatever n is) root of a #Power Function(the nth root of (x^m)).  In these expressions x>0 and m and n are intergers.  (Sorry if that was confusing but I couldn’t figure out how to make the square root sign.)  =(

Now you should be able to do all these things with #Power Functions.  I hope this helped anyone who was struggling with this stuff.  If your still hungry for more well you will just have to wait until next math class.  If you really need more help here’s a great video to check out! Power Functions    And if you just want to look at some more #dominoes here you go! Dominoes!      

#Dominoes  #Takingcareofbusiness  #PowerFunctions